Moody, who had worked with Mackay at Watford, has been replaced by with 23-year-old Kazakhstani Alisher Apsalyamov on an interim basis.
Apsalyamov is thought to be a friend of Cardiff owner Vincent Tan's son and appears to have no previous experience in football.
Friday's press conference ahead of Saturday's Premier League meeting with Chelsea was the first opportunity for Mackay to speak publicly on the issue.
But, after clarifying there had been no overspend in the summer transfer budget and that he had not been asked to resign, he refused to answer any further questions on the matter.
He said: "Iain is an absolute class act as far as I am concerned. He has done a very impressive job for the football club.
"In one of the periods of success at the football club, which we have been in over the last two and a half years, he has had a huge part to play in the restructuring of my squad.
"He helped bring in more than 20-odd players and he is someone who had a huge part to play in the current success, who will be missed by everyone who works for out football club.
"He leaves with my utmost gratitude and respect. I would say our loss will be someone's huge gain.
"One thing I would clarify, as I have read a few things over the past couple of weeks, is that we were absolutely under budget this summer. I want to clarify that. That is what I have to say about Iain.
"As far as I am concerned, again for clarification, under no circumstances was I asked to resign and at no time have I thought of resigning from the football club."
Mackay's agent had suggested the Scot would lift the lid on why Moody had left if the club did not provide an explanation, but the Scot did not do so.
However, he heaped praise on chairman Mehmet Dalman for the role he played in an extended board meeting on Monday where events of recent weeks were discussed in some detail.
"On Monday and over the last two weeks I have a lot of close contact with the new chairman Mehmet Dalman, who is a very impressive figure," said Mackay, who also refused to confirm whether he had had any dealings with Apsalyamov.
"He chaired the board meeting as though he had chaired a thousand in his sleep, he was very impressive.
"Over the last couple of weeks there has been a mutual respect between us and an understanding of what needs to be done to make the club strong again.
"I hope he plays a big part going forward as far as the club is concerned.
"Steve Borley is another member of the board who has been at the club and been a firm supporter of the club for a long time.
"He has been someone who has shown his mettle. He desperately wants the club to move forward and be successful. He is someone who needs to be at the fore of things.
If we are going to go forward from here Mehmet Dalman is someone I have a lot of trust in that the club will be in the hands of someone who has a very good business acumen.
"Over the last week he has helped draw a line under some of the things that have happened and try and make things better going forward. After that I would say I will not say anymore on this issue.
"Any other questions in terms of where that's concerned, you can ask or ask to discuss with the CEO (Simon Lim) down at the stadium.
"I want to protect the players. I want to talk about football, we have a game against Chelsea. I am the manager and my job is to look after the football side of the club so, with respect, that's all I will be saying about that."